If y+frac{d}{d x}(x y)=x(sin x+log x),then fi | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given equation is $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$.Expanding the derivative: $y + y + x \frac{dy}{dx} = x \sin x + x \log x$.$2y + x \f

Vedclass · Accepted Answer

$y=-\cos x+\frac{2}{x} \sin x+\frac{2}{x^2} \cos x+\frac{x}{3} \log x-\frac{x}{9}+\frac{c}{x^2}$,where $c$ is the constant of integration.

Vedclass · Answer

(C) Given equation is $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$.Expanding the derivative: $y + y + x \frac{dy}{dx} = x \sin x + x \log x$.$2y + x \frac{dy}{dx} = x \sin x + x \log x$.Dividing by $x$: $\frac{dy}{dx} + \frac{2}{x} y = \sin x + \log x$.This is a linear differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$,where $P(x) = \frac{2}{x}$ and $Q(x) = \sin x + \log x$.Integrating factor $IF = e^{\int P(x) dx} = e^{\int \frac{2}{x} dx} = e^{2 \log x} = x^2$.The solution is $y \cdot IF = \int Q(x) \cdot IF dx + c$.$y \cdot x^2 = \int x^2(\sin x + \log x) dx + c$.$y x^2 = \int x^2 \sin x dx + \int x^2 \log x dx + c$.Using integration by parts for $\int x^2 \sin x dx$: $-x^2 \cos x + 2x \sin x + 2 \cos x$.Using integration by parts for $\int x^2 \log x dx$: $\frac{x^3}{3} \log x - \frac{x^3}{9}$.So,$y x^2 = -x^2 \cos x + 2x \sin x + 2 \cos x + \frac{x^3}{3} \log x - \frac{x^3}{9} + c$.Dividing by $x^2$: $y = -\cos x + \frac{2 \sin x}{x} + \frac{2 \cos x}{x^2} + \frac{x}{3} \log x - \frac{x}{9} + \frac{c}{x^2}$.This matches option $C$.

If $y+\frac{d}{d x}(x y)=x(\sin x+\log x)$,then find $y$.

Similar Questions

Let $y=y(x)$ be the solution of the differential equation $x^{4}dy + (4x^{3}y + 2\sin x)dx = 0$,$x>0$,$y(\frac{\pi}{2})=0$. Then $\pi^{4}y(\frac{\pi}{3})$ is equal to:

If for the solution curve $y=f(x)$ of the differential equation $\frac{dy}{dx}+(\tan x)y=\frac{2+\sec x}{(1+2\sec x)^2}$,$x \in \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$,$f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$,then $f\left(\frac{\pi}{4}\right)$ is equal to:

Let $Y=Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the tangent line $Y-y=Y^{\prime}(x)(X-x)$ and the coordinate axes,where $(x, y)$ is any point on the curve,is always $\frac{-y^2}{2 Y^{\prime}(x)}+1$,where $Y^{\prime}(x) \neq 0$. If $Y(1)=1$,then $12 Y(2)$ equals

Let $y=y_{1}(x)$ and $y=y_{2}(x)$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$,with $y_{1}(0)=0$ and $y_{2}(0)=1$ respectively. Then,the number of points of intersection of $y=y_{1}(x)$ and $y=y_{2}(x)$ is.

The integrating factor of the differential equation $(2x + 3y^2) dy = y dx$ $(y > 0)$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module